UNIVERSIDAD NACIONAL DE INGENIERÍA

FACULTAD DE INGENIERIA CIVIL

CENTRO PERUANO JAPONES DE INVESTIGACIONES

SISMICAS Y MITIGACION DE DESASTRES

Investigations of Shear Walls Matthias Schmalzbauer

May 2011

General

In structural engineering, a shear wall is a wall composed of braced panels to counter the effects of lateral load acting on a structure. Wind and earthquake loads are the most common loads braced wall lines are designed to counteract.

A fast way analyzing a concrete wall for large eccentricities is the “Interaction Diagram”. The Diagram is a curve plot of points, where each point hast two ordinates. The first ordinate is the bending moment strength and the second one is the corresponding axial force. Both ordinates are linked with eccentricity. The shape of the curve can be defined by finding the ordinates of major points.

General calculation of P and M

c

dn

εc

εc εsu+( ) cεc

εc εsu+( ) dn⋅

fsi

fy

εsuεsi⋅ 4200

kgf

cm2

≤ 4200−kgf

cm2

≥

εsi

c di−( )c

εc⋅

β 0.85f¨c 200−( )

700.05⋅−

Fsi fsi Asi⋅

Cc 0.85 f´c⋅ B⋅ a⋅

a β c⋅P Cc

1

n

i

Fsi∑=

+

M CcH

2

a

2−

⋅

1

n

i

FsiH

2di−

⋅

∑

=

+

Boundary conditions:

0.85

0.65

β

fc

kgf

cm2

280 560

Diagram for finding β

Section

Deformation at the “Balanced condition”

Forces at the “Balanced Condition”

Finding c at “Balanced Condition”:

Note: Balanced Condition:

Tension failure = Compression failure

Deformation:

Forces:

Resulting Force:

(Balance of Forces)

q

(Moments around q)

Main points “Interaction Diagram”

Point 1: Axial compression at zero moment. Point 2: Maximum axial compression load permitted by code at zero eccentricity. Point 3: Maximum moment strength at the maximum axial compression load permitted by code. Point 4: Compression and moment strength at balanced conditions. Point 5: Moment strength at zero axial force.

-100

0

100

200

300

400

500

600

0 20 40 60 80 100 120 140 160

Ax

ial

Co

mp

ress

ion

P [

t]

Bending Moment M [t*m]

1

2

3

4

5

comp. control

tension control

balanced

Strain distribution corresponding to points on the "Interaction Diagram"

Any combination of P and M outside the envelope will cause failure

Strain distribution corresponding to points on the "Interaction Diagram"

Any combination of P and M outside the envelope will cause failure

Concrete failure

Steel failure

Strain distribution corresponding to points on the "Interaction Diagram"

Any combination of P and M outside the envelope will cause failure

Calculation

Shear-Wall Type A: Rectangular shape

(Calculation - Excel)

Material Definition: (Entered Data)

Geometric Data: (Entered Data)

Illustrated shape Actual arrangement

(β changes with change of f`c)

(Yield strength)

(Compressive strength)

(Strain concrete)

(Yield strain steel)

ΣA si

H B⋅( )

Asiπ4

φ i

8

2.54⋅ cm⋅

2

⋅

cB

εc

εc εsu+( ) dn⋅

aB β cB⋅

fsi

fy

εsuεsi⋅

Fsi fsi Asi⋅

Msi FsiH

2d−

⋅

M CcH

2

a

2−

⋅

1

n

i

FsiH

2di−

⋅

∑

=

+ P Cc

1

n

i

Fsi∑=

+

Cc 0.85 f´c⋅ B⋅ aB⋅

εsi

cB di−( )cB

εc⋅

Reinforcement:

Calculation Balanced-Point: (Point 4) Note: Getting more points by changing cB

Interaction Diagram Shear-Wall Type A: (Rectangular shape)

-100

0

100

200

300

400

500

600

0 20 40 60 80 100 120 140 160

Ax

ial

Fo

rce

P [

t]

Bending Moment M [t*m]

Pmax 0.85 f'c⋅ B H⋅

1

n

i

Asi∑=

−

⋅

1

n

i

Asi∑=

fy⋅+

Further Results with changed c: (Axial Force “P” and Bending Moment “M”)

Calculation: (Point 1) - Axial compression at zero moment

(Point 1)

(Point 4)

Shear-Wall Type B: Wall with two columns

(Calculation - Excel)

Material Definition: (Entered Data)

Geometric Data: (Entered Data)

Illustrated shape Actual arrangement

(β changes with change of f`c)

(Yield strain steel)

(Strain concrete)

(Yield strength)

(Compressive strength)

ΣA si

H B⋅( )

Asiπ4

φ i

8

2.54⋅ cm⋅

2

⋅

Pmax 0.85 f'c⋅ B H⋅

1

n

i

Asi∑=

−

⋅

1

n

i

Asi∑=

fy⋅+

Reinforcement:

Calculation: (Point 1) - Axial compression at zero moment

cB

εc

εc εsu+( ) dn⋅

aB β cB⋅

fsi

fy

εsuεsi⋅

Fsi fsi Asi⋅

Msi Fsi

2 H2⋅ H1+( )2

d−

⋅

Cc 0.85 f´c⋅ B2⋅ a⋅

P Cc

1

n

i

Fsi∑=

+

εsi

cB di−( )cB

εc⋅

M Cc

H1 2 H2⋅+( )2

a

2−

⋅

1

n

i

Fsi

2 H2⋅ H1+( )2

⋅ di−

∑

=

+

Calculation Balanced-Point: (Point 4) Note: Getting more points by changing cB

Interaction Diagram Shear-Wall Type B: (Wall with two columns)

-500

0

500

1000

1500

2000

0 100 200 300 400 500 600 700

Ax

ial

Fo

rce

P [

t]

Bending Moment M [t*m]

Further Results with changed c: (Axial Force “P” and Bending Moment “M”)

(Point 1)

(Point 4)

Calculation

Shear-Wall Type A: Rectangular shape

(Mathcad)

Material definitions

E-Module Concrete - (Kent and Park)

Concrete under multiaxial compressive stress state exhibits significant nonlinearity, which can be successfully represented by nonlinear constitutive models and is characterized by parabolic stress-strain relationship. Elastic limit strain and strain at cracking are limited to 0.2%.

Entered data:

fc 0.85 280⋅kgf

cm2

:=

fc 238kgf

cm2

⋅=

f εc( ) fc 2εc

0.002⋅

εc0.002

2

−

⋅ εc 0.002≤if

fc 1 100 εc 0.002−( )⋅−[ ]⋅ εc 0.002>if

:=

0 1 103−

× 2 103−

× 3 103−

× 4 103−

×0

1 107

×

2 107

×

3 107

×

f εc( )

εc

Stress-strain graphic for concrete (Kent and Park)

(Maximum stress of concrete)

0.85: Standarts

E-Module Steel - (Bilinear diagram with parable – Park and Paulay)

Steel is isotropic and homogeneous material exhibiting stress-strain relationship. While the ultimate limit strain in tension and that of compression are taken as 1% and 0.35%, respectively, elastic strain in steel in tension and compression are considered the same.

Entered data:

Definition: steel: Creep Deformation sh: Deformation hardening steel su: Ultimate strain of steel Es: E-Module steel. fy: Yield stress steel. fu: Steel final effort

fy 4200kgf

cm2

⋅:= fu 6300kgf

cm2

⋅:= Es 29000 1000⋅lbf

in2

⋅:= Es 1.999 1011× Pa=

t

fu

fy

30 q⋅ 1+( )2⋅ 60 q⋅− 1−

15 q2⋅

:=

fs εs( ) fy−t εs− εsh−( )⋅ 2+

60 εs− εsh−( )⋅ 2+εs− εsh−( ) 60 t−( )⋅

2 30 q⋅ 1+( )2⋅

+

⋅ εs εsh<if

fy− εsh− εs≤ εsteel−<if

Es εs⋅ εsteel− εs≤ εsteel≤if

fy εsteel εs≤ εsh≤if

fyt εs εsh−( )⋅ 2+

60 εs εsh−( )⋅ 2+εs εsh−( ) 60 t−( )⋅

2 30 q⋅ 1+( )2⋅

+

⋅ εs εsh>if

:=

εsteelfy

Es:= εsteel 2.06 10

3−×=

εsu εsh 0.14+:=

εsh 16 εsteel⋅ εsh 0.033=

εsu 0.173=

q εsu εsh−:= q 0.14=

t 105.986=

Deformation in flow:

General formula for steel:

Deformation at the hardening:

Last Deformation steel:

Calculations for hardening steel:

0.1− 0 0.11− 10

9×

5− 108

×

0

5 108

×

1 109

×

fs εs( )

εs

γ εc( )2 250 εc⋅−( )

2 3 500 εc⋅−( )⋅εc .002≤if

1

56−

12 107⋅

6

10

εc 2⋅+100

3εc 3⋅−

εc26−

3 104⋅

12

10

εc⋅+ 50 εc 2⋅−

⋅− εc 0.002>if

:=

α εc( )10

3

6

3 εc⋅ 500 εc 2⋅−( )⋅ εc .002≤if

26−3 10000⋅ εc⋅

50 εc⋅−12

10+ εc 0.002>if

:=

Stress-strain graphic for steel – (Park and Paulay)

Geometric data

Entered data:

d11

d12

d21

d22

dn

h 200 cm⋅:= f́ c 280kgf

cm2

:=b 10 cm⋅:=

fy 4200kgf

cm2

:=ε c 0.003:=

ε su 0.002:=

d

194cm

194cm

170cm

150cm

130cm

110cm

90cm

70cm

50cm

30cm

6cm

6cm

:=

φ 1 4:=φ 2 4:=

φ 3 3:=φ 4 3:=φ 5 3:=φ 6 3:=φ 7 3:=φ 8 3:=φ 9 3:=φ 10 3:=φ 11 4:=φ 12 4:=

dmax max d( ) 194 cm⋅=:=

(Compressive strength)

(Yield strain steel)

(Strain concrete)

(Strain steel)

(Number diameter steel)

Calculation Steel-Areas:

Asi

π4

φ 1

8

2.54⋅ cm⋅

2

⋅

π4

φ 2

8

2.54⋅ cm⋅

2

⋅

π4

φ 3

8

2.54⋅ cm⋅

2

⋅

π4

φ 4

8

2.54⋅ cm⋅

2

⋅

π4

φ 5

8

2.54⋅ cm⋅

2

⋅

π4

φ 6

8

2.54⋅ cm⋅

2

⋅

π4

φ 7

8

2.54⋅ cm⋅

2

⋅

π4

φ 8

8

2.54⋅ cm⋅

2

⋅

π4

φ 9

8

2.54⋅ cm⋅

2

⋅

π4

φ 10

8

2.54⋅ cm⋅

2

⋅

π4

φ 11

8

2.54⋅ cm⋅

2

⋅

π4

φ 12

8

2.54⋅ cm⋅

2

⋅

:= Asi

1.267

1.267

0.713

0.713

0.713

0.713

0.713

0.713

0.713

0.713

1.267

1.267

cm2⋅=

Calculation: (Point 1) - Axial compression at zero moment

Balanced point: (Point 4)

Pmax 0.85 f́ c⋅ b h⋅

0

11

i

Asi i∑=

−

⋅

0

11

i

Asi i∑=

fy⋅+

:=

P max 518.661 tonnef=

εsi

cB d0−

cBεc⋅

cB d1−

cBεc⋅

cB d2−

cBεc⋅

cB d3−

cBεc⋅

cB d4−

cBεc⋅

cB d5−

cBεc⋅

cB d6−

cBεc⋅

cB d7−

cBεc⋅

cB d8−

cBεc⋅

cB d9−

cBεc⋅

cB d10−

cBεc⋅

cB d11−

cBεc⋅

:=

cB dmax

εc

εc εsu+⋅

116.4 cm⋅=:=

fsi

fs εsi0

fs εsi1

fs εsi2

fs εsi3

fs εsi4

fs εsi5

fs εsi6

fs εsi7

fs εsi8

fs εsi9

fs εsi10

fs εsi11

3-4.078·10

3-4.078·10

3-2.817·10

3-1.766·10

-714.667

336.314

31.387·10

32.438·10

33.489·10

34.2·10

34.2·10

34.2·10

kgf

cm2

⋅=:=εsi

-3-2·10

-3-2·10

-3-1.381·10

-4-8.66·10

-4-3.505·10

-41.649·10

-46.804·10

-31.196·10

-31.711·10

-32.227·10

-32.845·10

-32.845·10

=

Note: Getting more points by changing cB

Interaction Diagram Mathcad Type A: (Rectangular s hape)

0 50 100 150

200−

0

200

400

600

Pi

Mi

P0 α εc( ) fc⋅ b⋅ cB⋅

0

11

i

fsi iAsi i⋅

∑

=

+:=

P0 210.852 tonnef⋅=

M0 α εc( ) fc⋅ b⋅ cB⋅h

2γ εc( ) cB⋅−

⋅

0

11

i

fsi iAsi i⋅

h

2

di−

⋅

∑

=

+:=

M0 109.933 tonnef m⋅⋅=

P

518.661

253.023

210.852

168.682

105.426

63.255

42.17

12.09−

:= M

0

107.698

109.933

104.093

80.196

54.173

38.134

0

:=

Further Results with changed c: (Axial Force “P” and Bending Moment “M”)

BP

tonnef tonnef m

(Balanced condition)

(Balanced condition)

Comparison of the two calculations:

The area of “Type B” is twice as big as from “Type A”. But the wall with two columns can absorb about three to four times larger moments and Axial Forces. Partly because more steel is installed in the wall (Type B)

Results Shear Wall Type A: Rectangular shape (Excel)

Results Shear Wall Type B: Wall with two columns (Excel)

Pmax: Same value because of the same area in both cases

Points (Fragile-, Ductile-Condition and Balanced Point): Different values with Kent and Park and Paulay because of a more strict method of calculation including by using the functions of the E-Modules.

References:

Reinforced Concrete (A Fundamental Approach) - Edward G. Nawy 5th Edition

Reinforced concrete structures R.Park and T.Paulay

Seismic design aids for nonlinear analysis of reinforced concrete structures

ACI-318

NTE E.060

Results Shear Wall Type A: Rectangular shape (Excel)

Results Shear Wall Type A: Rectangular shape (Mathcad)

(Pmax)

(Pmax)